My answer to your question there is, “No, but actually yes.”
No, because you have the 1D Reputation, which gives you an extra die when circling in the Rep’s purview. Clear-cut, easy-peasy.
Yes, because context is important. If you have a Rep for having saved the mayor’s son, you could probably squeeze another Advantage die or two from me when you are trying to Circle up the Mayor himself. (At least, for the first time or two. I’m a busy guy; what have you done for me lately?) On the hand, if I know the “tragic accident” that almost killed the mayor’s son was actually an assassin’s plot, and you try to Circle up an assassin, now you have an Infamous Reputation for this test because you’ve thwarted them before! The enmity clause features a desire for payback, an you’d wish you were the fastest dude in town! What might have been a simple failure or a requirement for a little quid-pro-quo (“Punch that guard and then run away so I can slip through the gate. You’re the fastest guy in town, right? You’ll get away easy.”) becomes a life-or-death Fight!
Reputations can also be factored into Relationship Aptitudes. You have an easier time forming a Relationship with the mayor if you’ve saved his son. As the fastest dude in town, you might build Relationships with craftsmen or shop-keepers (people who lean on couriers a lot).
In my estimation, there isn’t really a whole lot of “fluff” in Burning Wheel. The mechanics are so sensitive to context that just about anything in the fiction can be leveraged for or against a character.